HAL: hal-00597127, version 1

DOI: 10.1007/s10623-012-9704-4

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Designs, Codes and Cryptography (2012) nc

Linear codes using skew polynomials with automorphisms and derivations

Delphine Boucher 1, Félix Ulmer 1

(2012)

In this work the de nition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings whose multiplication is de ned using an automorphism and an inner derivation. This produces a more gen- eral class of codes which, in some cases, produce better distance bounds than skew module codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several ap- proaches to generalize Reed Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank matrix and derive families of MDS and MRD codes.

1:	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

Research team: Géométrie algébrique réelle

Subject

:

Mathematics/Information Theory Computer Science/Information Theory and Coding

Keyword(s): Error-correcting codes – Decoding – Finite fields – Skew polynomial rings

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From: Marie-Annick Guillemer <>

Submitted on: Tuesday, 31 May 2011 11:16:04

Updated on: Wednesday, 24 October 2012 14:27:42